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table of differences between and for
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(Data Structure)
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There are only three known solutions to Brocard's problem, and the near misses all seem to occur early on. Notice how, for example, 3! is just 3 shy of a square (compared to 1 shy of a square which is what Brocard's problem asks for). Still, the differences between a factorial and the next higher perfect square don't make
for a consistently ascending order sequence. For a few values of $n$ , (such as 4, 7, 10, 24, 26, 42, 117, 135) this difference is smaller than the previous difference. In general, however, the difference between a factorial and the next perfect square widens as $n$ gets larger.
The following table gives the square root of $n!$ to six decimal places, and then the difference between the factorial and the next higher square (obtained by taking the ceiling of the square root of $n!$ and squaring that integer).
| $n$ |
$\sqrt{n!}$ |
$\lceil \sqrt{n!} \rceil^2 - n!$ |
| 1 |
1.000000 |
0 |
| 2 |
1.414214 |
2 |
| 3 |
2.449489 |
3 |
| 4 |
4.898979 |
1 |
| 5 |
10.954451 |
1 |
| 6 |
26.832816 |
9 |
| 7 |
70.992957 |
1 |
| 8 |
200.798406 |
81 |
| 9 |
602.395219 |
729 |
| 10 |
1904.940944 |
225 |
| 11 |
6317.974359 |
324 |
| 12 |
21886.105181 |
39169 |
| 13 |
78911.474451 |
82944 |
| 14 |
295259.701280 |
176400 |
| 15 |
1143535.905864 |
215296 |
| 16 |
4574143.623456 |
3444736 |
| 17 |
18859677.306253 |
26167684 |
| 18 |
80014834.285449 |
114349225 |
| 19 |
348776576.634429 |
255004929 |
| 20 |
1559776268.628498 |
1158920361 |
| 21 |
7147792818.185865 |
11638526761 |
| 22 |
33526120082.371712 |
42128246889 |
| 23 |
160785623545.405884 |
191052974116 |
| 24 |
787685471322.938354 |
97216010329 |
| 25 |
3938427356614.691406 |
2430400258225 |
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Cross-references: integer, square root of 3, ceiling, decimal places, square root of 2, sequence, ascending order, perfect square, factorial, differences, square, near, Brocard's problem, solutions
This is version 2 of table of differences between and for , born on 2008-07-01, modified 2008-07-02.
Object id is 10735, canonical name is TableOfDifferencesBetweenLceilSqrtnRceil2AndNFor0N26.
Accessed 500 times total.
Classification:
| AMS MSC: | 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas) |
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Pending Errata and Addenda
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